In the black-box model, promise problems are the only ones that admit a quantum exponential speedup over the best classical algorithm in terms of query complexity. The most prominent example of this is the Deutsch-Jozsa algorithm. More recently, Wim van Dam put forward an algorithm for unstructured problems (ie, those without a promise). We consider the Deutsch-Jozsa algorithm with a less restr...

Longprt, L. and A.L. Selman, Hard promise problems and nonuniform complexity, Theoretical Computer Science 115 (1993) 277-290. For every recursive set A, let PP-A denote the following promise problem: input x and y promise (u~A)@(ysA) property XE A. We show that if L is a solution of PP-A, then A~p~/Poly. From this result, it follows that if A is <F-hard for NP, then all solutions of PP-A are h...